>>380 補足
>http://www-dimat.unipv.it/giulio/linkedmaterial/af/notaHamel.pdf
>Hamel basis and additive functions GIULIO SCHIMPERNA June 26, 2013

Hamel basisが分からないので、これ読み返していた（いまでもまだ”もや”としているが）
P1より
Existence of Hamel basis
Definition 1.
　 Let V be a vector space over a field K.
　 We say that B is a Hamel basis in V if B is linearly independent and every vector v ∈ V can be obtained as a linear combination of vectors from B.
　 This is equivalent to the condition that every x ∈ V can be written in precisely one way as
把ixi　（i∈F）
where F is finite, ci ∈ K and xi ∈ B for each i ∈ F.
　 It is also easy to see that for any vector space W and any map g : B → W
there exists exactly one linear map f : V → W such that f|B = g.

Theorem 1.
　 Let V be a vector space over K. Let A be a linearly independent subset of V .
　 Then there exist a Hamel basis B of V such that A ⊆ B. (Any linearly independent set is contained in a basis.)
Proof. Zorn's lemma.

Corollary 1.
　 Every vector space has a Hamel basis.
Proof. For V = {0} we have a basis B = φ.
If V ≠ {0}, we can take any non-zero element x ∈ V and use Theorem 1 for A = {x}.
In some cases we are able to write down a basis explicitly, for example infinitely-dimensional space or in the following example.
However, the claim that a Hamel basis exists for each vector space over any field already implies AC (see [HR, Form 1A]).

Example 1. Let c00 be the space of all real sequences which have only
finitely many non-zero terms.
Then {e(i); i ∈ N}, where the sequence e(i) is given by e(i)n = δin, is a Hamel basis of this space.